Journal of Computational Information Systems 11: 15 (2015) 5347–5352Available at http://www.Jofcis.com

Decomposition and Reconstruction Algorithms for

Framelet Packets ⋆

Dayong LU∗, Meiyu XU

Institute of Applied Mathematics, College of Mathematics and Information Science, Henan University,Kaifeng 475001, China

Abstract

Wavelet packets based on orthonormal wavelet bases have been well studied in theory and applications,since they can provide adaptive choice from a library of wavelet bases for a wide range of practicallyoriented tasks. But the study of wavelet frame packet have been less involved. In this paper, we give thedecomposition and reconstruction algorithms for framelet packets constructed from the unitary extensiongiven by Ron and Shen.

Keywords: Framelet Packets; Extension Principles; Fast Algorithms; Wavelet Frames

1 Introduction

Wavelet frames are nowadays indispensabel as a multiscale system in the applications of redundantdyadic wavelet systems, since they provide the same decomposition and reconstruction formula asorthonormal wavelet bases. Of all the wavelet frames, tight wavelet frames are the easiest to use.Tight wavelet frames are different from orthonormal wavelet bases in one important respect; theyare (in general) redundant systems but with the same fundamental structure as wavelet systems.To mention only a few references on tight wavelet frames, the reader is referred to [1-3].

However, wavelet frames provide poor frequency localization in applications. To overcome thisdisadvantage, the concept of wavelet frames must be generalized to include a library of waveletframes, called framelet packets or wavelet frame packets.

The original idea of wavelet packets were introduced by Coifman, Meyer, and Wickerhauser in[4, 5]. But the theory itself is worthy of further study. Some developments in the wavelet packetstheory should be mentioned, such as multiwavelet packets [6] on Rd, the non-tensor-productversion [7] of wavelet packets on Rd, the nonorthogonal version of wavelet packets [8] on R1,the wavelet frame packets [9] on R1 and the higher dimensional version of wavelet frame packets

⋆The work is supported by the Natural Science Foundation for the Education Department of Henan Provinceof China (No. 13A110072), the Natural Science Foundation of Henan Province (No. 122300410381), and HenanUniversity Natural Science Foundation (No. 2011YBZR001).

∗Corresponding author.Email address: dayonglu@163.com (Dayong LU).

1553–9105 / Copyright © 2015 Binary Information PressDOI: 10.12733/jcis11919August 1, 2015

5348 D. Lu et al. /Journal of Computational Information Systems 11: 15 (2015) 5347–5352

[10] on Rd. Recently, using the so-called splitting trick given by Daubechies [11], Lu and Fanin [1] constructed a class of tight framelet packets with 2Id-dilation for L2(Rd) from the unitaryextension principles given by Ron and Shen in [2]. In this paper, we consider the decompositionand reconstruction algorithms for framelet packets constructed in [1].

2 Preliminaries

We begin by introducing some notation and definitions we shall use.

H denotes a separable Hilbert space with inner product ⟨·, ·⟩ and norm ∥x∥ = ⟨x, x⟩ 12 for each

x ∈ H. Let J be a numerable index set. A countable system {ϕj}j∈J in H is called a frame for Hif there exist constants A and B, 0 < A ≤ B <∞, such that

A∥x∥2 ≤∑j∈J

|⟨x, ϕj⟩|2 ≤ B∥x∥2 (2.1)

holds for all x ∈ H. The greatest possible such A is the lower frame bound and the least possiblesuch B is the upper frame bound. If A = B, then the frame is called a tight frame.

Define the Fourier transform f of f ∈ L1(R) ∩ L2(R) by f(ξ) =∫R f(x)e

−ixξdx.

Translation by y ∈ R is denoted by Ty, i.e., if f : R → C is a function, then Tyf : R → C isthe function defined by (Tyf)(t) = f(t − y). Further, ∀f ∈ L2(R), the unitary dyadic dilation

operator D id defined on L2(R) as (Df)(x) =√2f(2x), and, hence, (Djf)(x) = 2

j2f(2jx) for all

j ∈ Z.In the following we shall briefly describe how to construct multiresolution analysis (MRA)-

based tight wavelet frames through so-called extension principles, see [3, 4]. We refer the readerto [3] for a more detailed discussion of MRA-based wavelet frames.

Let τ = {τ0, τ1, . . . , τL} be a sequence of 2πZ-periodic essentially bounded functions. Assumethat τ0 generates the refinable function ϕ(2ξ) = τ0(ξ)ϕ(ξ) satisfying

limξ→0

ϕ(ξ) = 1 and∑k∈Z

|ϕ(ξ + 2kπ)|2 ≤ B2 for some B. (2.2)

We associate the wavelets to τ as follows

ψl(2ξ) = τl(ξ)ϕ(ξ), l = 1, 2, . . . , L. (2.3)

We often write Ψ = {ψ1, ψ2, . . . , ψL}. The spectrum σ(ϕ) associated to ϕ is defined as

σ(ϕ) = {ξ ∈ [−π, π] : ϕ(ξ + 2kπ) = 0, for some k ∈ Z}. (2.4)

Let Ψ be a finite subset of L2(R). The dyadic wavelet system generated by Ψ is the family

X(Ψ) = {DjTkψ : ψ ∈ Ψ; j, k ∈ Z}. (2.5)

The following theorem proved in [3] is the main tool to create tight wavelet systems, the theoremis called the Unitary Extension Principle (UEP).

D. Lu et al. /Journal of Computational Information Systems 11: 15 (2015) 5347–5352 5349

Proposition 2.1 (UEP) Let τ be the combined mask of an MRA that satisfies the aboveassumptions. If ξ ∈ σ(ϕ), and if ν ∈ {0, π} is such that ξ + ν ∈ σ(ϕ), then

τ0(ξ)τ0(ξ + ν) +L∑l=1

τl(ξ)τl(ξ + ν) =

1, if ν = 0,

0, otherwise.

Then the wavelet system X(Ψ) defined by τ is a tight wavelet frame.

Remark 2.2 In many (most) interesting cases the spectrum σ(ϕ) is equal to [−π, π]. Forexample, if the integer translates of the scaling functions ϕ generates a Riesz sequence, this is thecase.

A wavelet system X(Ψ) is said to be MRA-based if it is generated by OEP or UEP. Theelements in X(Ψ) are called framelets. The collection Ψ is called the mother wavelet set, andthe elements in Ψ are called mother wavelets. We call τ0 the refinement mask and functions τl,l = 1, 2, . . . , L, wavelet masks. We call the sequence τ = {τ0, τ1, . . . , τL} the combined mask ofthe MRA.

3 Basic Framelet Packets and Their Fast Algorithms

Suppose Ψ = {ψ1, ψ2, . . . , ψL} is a tight frame generated by UEP associated with the refinablefunction ϕ and the combined mask τ = {τ0, τ1, . . . , τL}.Let ω0 = ϕ. The basic framelet packets ωn(x), n = 0, 1, . . ., associated with the refinable

function ϕ are defined recursively by

ωn(L+1)+l(2ξ) = τl(ξ)ωn(ξ), l = 0, 1, . . . , L. (3.1)

When n = 0 and l = 0 in (3.1) we obtain the refinable function ϕ by its Fourier transform

ω0(2ξ) = τ0(ξ)ω0(ξ). (3.2)

When n = 0 and l ∈ {1, 2, . . . , L} we deduce that

ωl(2ξ) = τl(ξ)ω0(ξ), (3.3)

which shows that ωl = ψl, l = 1, 2, . . . , L.

An important difference between wavelet frames and framelet packets is the decompositionstructure. We can depict these wavelet frame decompositions when L = 2 as given in Fig. 1, butframelet packets decompositions with compactly supported tight wavelet frames when L = 2 asgiven in Fig. 2.

Define the subspaces of L2(R) by

Unj := span{DjTkωn : k ∈ Z}, j ∈ Z, n = 0, 1, 2, . . . . (3.4)

We have the following relationships about the subspaces Unj , j ∈ Z and n = 0, 1, 2, . . ..

Theorem 3.1 [1] For n = 0, 1, 2, . . . we have

Unj+1 = U

n(L+1)j + U

n(L+1)+1j + · · ·+ U

n(L+1)+Lj , j ∈ Z, (3.5)

5350 D. Lu et al. /Journal of Computational Information Systems 11: 15 (2015) 5347–5352

0

���������

�� ��===

====

0

���������

�� ��===

====

1 2

0 1 2

Fig. 1: Wavelet frame de-composition

0

uujjjjjjjj

jjjjjjjj

jjjjj

��))TTT

TTTTTTTT

TTTTTTTT

TT

0

���������

�� ��===

====

1

���������

�� ��===

====

2

���������

�� ��===

====

0 1 2 0 1 2 0 1 2

Fig. 2: Framelet packet decomposition

where Unj is defined by (3.4).

Associated with the sequence of subspaces {Unj } we have the projections of L2(R) onto Un

j

given by

P nj f =

∑k∈Z

⟨f,DjTkωn⟩DjTkωn ∀f ∈ L2(R).

We can easily getL∑l=0

Pn(L+1)+lj f = P n

j+1f ∀f ∈ L2(R). (3.6)

From Theorem 3.1, we know that f ∈ L2(R) can be written as

f =∑k∈Z

⟨f,DjTkωn⟩DjTkωn. (3.7)

Thus, we have the coefficients

cn,jk = ⟨f,DjTkωn⟩, j, k ∈ Z, n = 0, 1, 2, . . . , (3.8)

and what we want to do is to decompose the sequence

cn,j = {cn,jk : k ∈ Z} (3.9)

which belongs to l2(Z).We now continue with the decomposition algorithm. This is achieved by the combined mask

τ = {τ0, τ1, . . . , τL} and (3.1). For convenience, we write τl as

τl(ξ) =∑m∈Z

αlme

imξ, l = 0, 1, 2, . . . , L (3.10)

So we have1

2ωn(L+1)+l(

x

2) =

∑m∈Z

αlmωn(x+m), l = 0, 1, 2, . . . , L. (3.11)

Hence, for l = 0, 1, 2, . . . , L,

Dj−1Tkωn(L+1)+l(x) = 2j−12 ωn(L+1)+l(2

j−1x− k)

= 2j+12

∑m∈Z

αlmωn(2

jx− 2k +m).

D. Lu et al. /Journal of Computational Information Systems 11: 15 (2015) 5347–5352 5351

That is,

Dj−1Tkωn(L+1)+l(x) =√2∑m∈Z

αlmD

jT2k−mωn(x), j, k ∈ Z, l ∈ {0, 1, 2, . . . , L}. (3.12)

It follows that, for all l = 0, 1, 2, · · · , L,

cn(L+1)+l,j−1k = ⟨f,Dj−1Tkωn(L+1)+l⟩ = ⟨f,

√2∑m∈Z

αlmD

jT2k−mωn⟩

=√2∑m∈Z

αlm⟨f,DjT2k−mωn⟩ =

√2∑m∈Z

αlmc

n,j2k−m.

(3.13)

This shows that the coefficients cn(L+1)+l,j−1 of the lowest resolution Un(L+1)+lj−1 can be obtained

from the coefficients cn,j of the Unj and the filter coefficients. For n and j fixed, the right-hand

side of (3.11) is the convolution of the sequences

αl = {√2αl

m} and cn,j = {cn,jm },

followed by retaining only the convolution entries that appear in the even places. The process toobtain the decomposition algorithm when L = 2 is given in Fig. 3.

cn,j

xxrrrrrr

rrrr

�� &&NNNNN

NNNNNN

c3n,j−1 c3n+1,j−1 c3n+2,j−1

Fig. 3: Decomposition algorithm for framelet packets

In the above we get the fast framelet packet decomposition algorithm, and we now treat theproblem of reconstruction cn,j from the sequences cn(L+1)+l,j−1, l = 0, 1, . . . , L.

Let us denote by

Cn,j(ξ) =∑k∈Z

cn,jk eikξ

the fourier series of cn,j for all j ∈ Z and n = 0, 1, 2, . . .. With some (but not much) effort, oneshows that (3.13) can be written on the frequency domain as

√2Cn(L+1)+l,j−1(ξ) = τl(

ξ

2)Cn,j(

ξ

2) + τl(

ξ

2+ π)Cn,j(

ξ

2+ π). (3.14)

Now substitute 2ξ for ξ, and then multiply each side of the Eq. (3.14) by τl(ξ), and sum over alll. Then

√2

L∑l=0

τl(ξ)Cn(L+1)+l,j−1(2ξ) =

L∑l=0

|τl(ξ)|2Cn,j(ξ) +L∑l=0

τl(ξ)τl(ξ + π)Cn,j(ξ + π). (3.15)

By the filter conditions we have

√2

L∑l=0

τl(ξ)Cn(L+1)+l,j−1(2ξ) = Cn,j(ξ), (3.16)

5352 D. Lu et al. /Journal of Computational Information Systems 11: 15 (2015) 5347–5352

which implies that

cn,jk =√2

L∑l=0

∑m∈Z

αlk−2mc

n(L+1)+l,j−1 (3.17)

hold for all j, k ∈ Z and n = 0, 1, 2, . . .. Then Eq. (3.17) is the so-called fast framelet packetreconstruction algorithms.

Eq. (3.17) allows us to add the sequences cn(L+1)+l,j−1 to obtain cn,j, and the reconstructionalgorithm when L = 2 given in Fig. 4.

c3n,j−1

&&LLLLL

LLLLLL

c3n+1,j−1

��

c3n+2,j−1

xxpppppp

pppppp

cn,j

Fig. 4: Reconstruction algorithm for framelet packets

Acknowledgement

We would like to thank the referees for their helpful comments and suggestions.

References

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